Democratic senators Heidi Heitkamp and
Joe Manchin may actually vote in favour of the confirmation. So the GOP potentially has a lot to gain by pleasing those two senators, as well as its own potential dissenters.

In theory (if they have votes, which as Martin's answer notes, they may not necessarily have), they can.

In practice, they have to worry about the politics (specifically, optics) of the thing, and how it will affect voter sentiment and therefore subsequent elections, both 2018 midterms as well as 2020 and on.

If they force the vote, and they have enough votes in the Senate to push this through, it will likely be hugely unpopular (Kavanough is iirc polling at -5 already; and swing voters are likely to be displeased); so Senate Majority leader - who as per 538 reporting, wasn't happy with the candidate being nominated in the first place - is not anxious to take on this gamble.

If they really worried about optics, they could just (persuade the president to) nominate and fast track another judge, right? Or are you aware of a reason they really need to nominate this judge?
– JJJOct 1 at 11:47

2

@JJJ - Separation of Powers. They can TRY to persuade the President. Whether they will be successful is a different story. Plus, they may be worried they are out of time to switch nominees (rightly or wrongly).
– user4012Oct 1 at 11:51

1

@JJJ It is true that they could hastily nominate and cram through another candidate before the midterms. The problem is that doing so has similar downsides to forcing a vote on Kavanaugh; it would almost certainly be viewed by the American public as valuing a "win for their side" over the proper and careful consideration of a lifetime appointment to the highest court in the land.
– BradCOct 1 at 15:49

@BradC they have till the new senate is installed (I think that's not directly after the midterms). As for putting forward another candidate, they could easily explain that by saying they have lost confidence in the current candidate (which given the situation would carry well with the public).
– JJJOct 1 at 15:54

@user4012 separation of powers means they can reject Kavanaugh and there's nothing Trump could do about it except refusing to nominate someone else until the end of his term.
– JonathanReezOct 1 at 16:24